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If there are k positions in a sequence, and n1 distinguishable elements can occupy position 1, n2 can occupy position 2, ... and nk can occupy position k, the number of distinct sequences of k elements each is given by

N (k | n1, n2, …, nk) = n1n2…nk(B.1)

For example, if a design involves three parameter φ1, φ2, φ3, and there are, respectively, 2, 3, 4 values of these parameters, the number of feasible design is

N (3 | 2, 3, 4) = (2) (3) (4) = 24

In a set of n distinct elements, the number of k-element ordered sequences, or arrangements, is:

(B.2)

For example, if no digits are repeated, the number of four-digit figures is (10)4 = (10) (9) (8) (7) = 5040, whereas, if the digits can be repeated the number of four-digit figures would be (10)4 = 10,000—the latter case would include 0000 as one of the figures.

Combination

In a set of n distinguishable elements, the number of possible subsets of k different elements each (regardless of order) is given by the binomial coefficient

(B.3)

Equation B.3 is defined only for k ≤ n. Using Eq. B.2 for (n)k, we have

(B.3a)

For example, the number of possible samples of four cards each in a deck is: